Classical ideals theory of maximal subrings in non-commutative rings

TL;DR

This paper investigates the relationships between various important ideals in maximal subrings of non-commutative rings, providing conditions under which these ideals coincide or relate in specific ways, especially in reduced or Artinian rings.

Contribution

It establishes new connections between ideals of maximal subrings and their over-rings, especially in reduced and Artinian cases, expanding understanding of ideal structures in non-commutative ring theory.

Findings

If $T$ is reduced, either $Z({}_RT)=0$ or it is minimal, with $T=Rigoplus Z({}_RT)$.

When $T=Rigoplus I$, the relations between radicals and socles are fully characterized.

In left Artinian cases, specific ideal relations between $R$ and $T$ are determined.

Abstract

Let $R$ be a maximal subring of a ring $T$. In this paper we study relation between some important ideals in the ring extension $R\subseteq T$. In fact, we would like to find some relation between $Nil_*(R)$ and $Nil_*(T)$, $Nil^*(R)$ and $Nil^*(T)$, $J(R)$ and $J(T)$, $Soc({}_RR)$ and $Soc({}_RT)$, and finally $Z({}_RR)$ and $Z({}_RT)$; especially, in certain cases, for example when $T$ is a reduced ring, $R$ (or $T$) is a left Artinian ring, or $R$ is a certain maximal subring of $T$. We show that either $Soc({}_RR)=Soc({}_RT)$ or $(R:T)_r$ (the greatest right ideal of $T$ which is contained in $R$) is a left primitive ideal of $R$. We prove that if $T$ is a reduced ring, then either $Z({}_RT)=0$ or $Z({}_RT)$ is a minimal ideal of $T$, $T=R\oplus Z({}_RT)$, and $(R:T)=(R:T)_l=(R:T)_r=ann_R(Z({}_RT))$. If $T=R\oplus I$, where $I$ is an ideal of $T$, then we completely determine relation between Jacobson radicals, lower nilradicals, upper nilradicals, socle and singular ideals of $R$ and $T$. Finally, we study the relation between previous ideals of $R$ and $T$ when either $R$ or $T$ is a left Artinian ring.